PHYS366: Statistical Methods in Astrophysics

Lesson 2: Understanding From Data

Goals for this session:

• Be able to describe the Bayesian view of models, parameters and uncertainty

• Know how to set up and perform simple Bayesian inferences, including the assignment of probability distributions

• Be able to draw simple PGMs and understand their connection to probability expressions

Related reading:

• MacKay Chapter 2, sections 2.1, 2.1, 2.3, and Chapter 3, sections 3.1, 3.2
• Ivezic Chapter 3, sections 3.1, 3.3 and Chapter 5, sections 5.1, 5.2, 5.3

Generative Models and Posterior Inferences

A good way to start understanding a dataset is to try to recreate it.

• Last time we made our first generative model, for SDSS galaxy sizes. It went past quickly, so let's revisit it, and look at how we sampled from some probability distributions.

• Let's make another forward model, this time with the X-ray Image dataset

• A short diversion about models in science

Now, let's look at the inverse problem, and start learning model parameters from data. We already started using probability distributions for random variables, but now we'll need to think a bit more carefully.

• Inferring cluster model parameters from the X-ray Image

Example: The Cepheid Period-Luminosity Relation

• Let's get some more practice in setting up a Bayesian inference.

• This boils down to two tasks: 1) assigning probability distributions (for both the data and the parameters) and 2) doing integrals.

• Let's work through a simple model for some of our Cepheid data.

Assigning Priors

In the previous examples, we assigned uninformative prior PDFs for our model parameters, without much thought.

• Let's look at some general principles, and some health warnings, about priors

Recap

• What did we learn about inference?